PIERRE DE FERMAT

Yogita Chellani

Term Paper, History of Mathematics, Rutgers

The French mathematician Pierre de Fermat(1601-1665) was
possibly the most productive mathematician of his era, making many
contributions, some of which were to calculus, number theory, and the
law of refraction. We will survey those contributions here, paying
particular attention to his work in number theory.

The following account of Fermat's background is taken from
Mahoney's book, The Mathematical Career of Pierre de Fermat. Pierre
de Fermat was born on August 17, 1960, in Beaumont-de-Lomagne, a small
town near Toulouse in the south part of France, near the border with
Spain. His father, Dominique Fermat, was a wealthy leather merchant
who held the position of "second consul" of Beaumont-de-Lomagne, a
governmental position similar to the position of mayor in our time.
His mother, Claire, née de Long, was the daughter of a prominent
family. Fermat had a brother, Clément, and two sisters, Louise and
Marie.

While relatively little is known of Fermat's early education, it
is known that he was of Basque origin and received his primary and
secondary education at the monastery of Grandsl ve, run by the
Cordeliers (Franciscans), in Beaumont-de-Lomagne. For his advanced
studies he first attended the University of Toulouse before moving to
Bordeaux in the second half of the 1620's. In Bordeaux (1629) Fermat
began his first serious mathematical researches, where he gave a copy
of his restoration of Appollonius's Plane Loci to one of the
mathematicians there. In Bordeaux he contacted Beaugrand and during
this time he produced work on maxima and minima. He gave his work to
Etienne de'Espagnet, who shared mathematical interests with Fermat.

From Bordeaux Fermat went to study at the University of Law at
Orléans. On May 1, 1631 he received the degree of Bachelor of Civil
Laws. Fermat's choice of a legal career was natural and typical of
his time, for his father's wealth and his mother's famil y background.
To be in this career was an avenue to a higher social status and
political power. After graduating he purchased the office of
councillor at the parliament in Toulouse. Soon after that he acquired
a wife. She was his cousin fourth removed, Lo uise de Long. He gave
a dowry of 12,000 livres, which was not a problem for the young
lawyer. Soon after, he served in the local parliament and became
councillor, or legislator. His entire family, now including his
father-in-law, were members of the upper
class. Mahoney tells us how this affected his social status as well.

"Fermat's offices made him a member of that social
class also and entitled him to add the "de" to his
name, which he did from 1631 on."
(Mahoney, p.16)

The "de" is the mark of nobility in France.

Very little is known about Fermat's private life. He had five
children, Clément-Samuel, Jean, Claire, Catherine, and Louise.
Clément-Samuel was the oldest and closest to Fermat. He may have
shared many mathematical interests with Fermat. Clément-Samuel
eventually inherited his father's office of councillor.

For the remainder of his life Fermat lived in Toulouse, but he
also worked in his hometown of Beaumont-de-Lomagne, and the nearby
town of Castres. First he worked in the lower chamber of Parliament,
but then in 1638 he was appointed to the higher chamber , and finally
in 1652 he was promoted to the highest level in the criminal court.
This position was usually given to people of seniority, but since the
plague had struck in the early 1650's, many of the older men had died.
Fermat himself was struck down w ith the plague. In 1653 his death
was wrongly reported; Fermat had survived. This account of Fermat's
background and life was taken from [Mahoney, pp. 15-17].

Fermat contributed to the development of the calculus through his work on the properties of curves. He found the areas bounded by these curves, though a summation process. As Bell states,
"The creators of calculus, including Fermat, relied on geometric and physical(mostly kinematical and dynamical) intuition to get them ahead: they looked at what passed in their imaginations for the graph of a
continuous curve..." (Bell, p.59)

We now call this process integral calculus. It is ironic that Fermat
did not see what we now call the Fundamental Theorem of Calculus.
However, his work on this subject was an aid to developing the
differential calculus. In addition to his contribution t o calculus,
Fermat contributed to the law of refraction. He had a disagreement
with the philosopher and amateur mathematician, René Descartes.
"According to [Fermat's] principle, if a ray of light passes from a
point A to another point B, being reflected and refracted(refracted;
that is, bent, as in passing from air to water, or through a jelly of
variable density) in any manner during the passa ge, the path which it
must take can be calculated- all its twistings and turnings due to the
refraction, and all its dodgings back and forth due to reflections-
from the single requirement that the time taken to pass from A to B
shall be an extreme." (Bel l, p.63).

Descartes was trying to justify the sine law (Snell's law) by saying
that light travels more rapidly in the denser of the two media
involved in the refraction. Twenty years later Fermat realized that
this appeared to be in conflict with the Aristotelian view that nature
always chooses the shortest path.

Fermat applied his method of maxima and minima and made the
assumption that light travels less rapidly in the medium. He showed
that the law of refraction is consistent with the principle of least
time. "From this principle Fermat deduced the familiar laws of
reflection and refraction: the angle of reflection; the sine of the
angle of incidence(in refraction)is a constant number times the sine
of the angle of refraction in passing from one medium to anot her."
(Bell, p.63).

Finally I will discuss Fermat's contribution to number theory in
greater detail than his other contributions, as Bell does: "We begin
with a famous statement Fermat made about prime numbers." Bell first
reminds us that: "A positive
prime number, or briefly a prime number is any number greater than 1
which has as its divisor(without remainder) only 1 and the number
itself; for example 2,3,5,7,13,17, are primes and so are 257,65537"
(Bell, p.65)

He then explains that Fermat observed that the
numbers 3,5,17,257,65537, all belong to one sequence and can be
generated by one simple process: 3=2+1,
5=2^2+1,
17=2^4+1,
257=2^8+1,
65537=2^(16)+1.
So thus the sequence seen here is 2^(2^n)+1, for n=1 to infinity.
Now if we wanted to check if one of these
numbers is prime it would not be an easy process, unless you follow
Fermat, who was claiming that all numbers of the previous sequence
are prime. But Fermat was mistaken; he just guessed and did not prove
his idea. Euler, a century later, showed that 232 +1 has 641 as a
factor. So Fermat was wrong and we still do not know whether there
are any primes among the Fermat numbers,
for n>4.

Although Fermat made a mistake, through his work on numbers he
discovered that every prime number of the form 4n+1 is expressible as
the sum of two squares. However, Fermat as in almost all his
mathematical work left no written proof of this theorem. He did
however write a letter to his friend and mathematician, Carcavi. In
this letter he included how he proved this theorem. Bell tells us
that in his letter he wrote, "The course of my reasoning in
affirmative propositions is such: if an arbitrarily chosen prime of
the form 4n+1 is not a sum of two squares, [I prove that] there will
be another of the same nature, less than the one chosen, and
[therefore] next a third s till less and so on." (Bell, p.70).

So Fermat comes to the number 5 which is the least of all these
numbers. He sees that then 5 is not a sum of two squares. However,
it is. Then Fermat says in the letter according to Bell, "Therefore
we must infer by a reductio ad absurdum that all numbers of the form
4n+1 are sums of two squares." (Bell, p.70).

This causes confusion. Fermat is not explicit as to how he proved
his statement. We see that Fermat is using a device that he called
the method of "infinite descent". This is known as an inverted form
of reasoning by recurrence or mathematical induction. A more
important result derived from this is what is now known as Fermat's
Lesser Theorem. The theorem is that if p is a prime number and is a
is any positive integer, then ap - a is divisible by p. Once again no
proof was given. In this case Gotfried Lei bniz, the 17th Century
German mathematician and philosopher, and Leonhard Euler, the 18th
Century Swiss mathematician, provided proofs.

Finally, I will discuss briefly Fermat's last discovery, Fermat's
Last Theorem. While he was reading a copy of Diophantus' Arithmetica,
Fermat made a marginal note that remained unsolved until very
recently. He read the eighth problem in Dipohantus's boo k, which
asks for the solution in rational numbers of the equation x2 + y2= a2.
According to Bell Fermat says, "On the contrary, it is impossible to
separate a cube into two cubes, a fourth power into two fourth powers,
or generally, any power above the second into two powers of the same
degree: I have discovered a truly marvelous demonstration [of this
general theorem] which this margin is too narrow to contain" (Bell,
p.71).

To restate this, Fermat is saying that no nonzero whole numbers or
fractions exists with xn + yn = an, if n is a whole number greater
than 2. However, Fermat did not live to prove what he had in mind.
This theorem has puzzled mathematicians for many years
until recently, and is still remembered as Fermat's Last Theorem.

In conclusion, Pierre de Fermat has been called the greatest French mathematician of the seventeenth century (Eves,p.354). We have seen his contributions to calculus, the law of refraction, and most importantly to number theory. Fermat died on January 1
2, 1665 in Castres, France. Unfortunately Fermat's influence was not very great, because he was reluctant to publish his work. However, he is still remembered as a very great mathematician.